Ecosystem Management with Structural Dynamics
Rui Pedro Mota1,, Tiago Domingos, Environment and Energy Section, DEM, Instituto Superior Técnico
Abstract
The main objective of this work is to use an operational form of the concept of self-organisation in the context
of optimal ecosystem management. The problem we solve in this work is: what are the optimal management
policies for a self-organizing ecosystem along ecological succession? For this matter we develop a
bioeconomic model of an extractive economy exploiting an ecosystem with self-organisation along ecological
succession. We use an ecological production function that allows for feedbacks on ecosystem carrying
capacity. A dynamic carrying capacity is interpreted as accounting for the ecosystem’s structure or selforganisation. The ecosystem may, consequently, have three modes of growth: pure compensation, depensation
and critical depensation. This, in particular, implies that this model predicts the existence of minimum viable
ecosystem biomass for some parameters sets. The main conclusion is that the self-organization capability of
ecosystems has an influence on the optimal management policies. Accordingly, for pure compensation there is
one steady state for the bioeconomic system, and for all initial states the system converges to it. For
depensation and critical depensation there are two steady states, one stable and one unstable. In these cases,
there exists a possibility of optimal extinction. The interesting effect in an ecosystem with depensation and
critical depensation is that there is history dependence of the bioeconomic system. So, small changes in initial
states near the unstable steady state may have dramatically different long-term outcomes for the bioeconomic
system. The system may approach a positive steady state or extinction.
Keywords: Ecosystem management, self-organization, optimal policies.
1. Introduction
In this paper we address the problem of how to sustainably manage a self-organizing ecosystem that provides
ecological flows to a society. The problem is very easy to understand and has for a long time been faced by
humans. The harvesting of biological resources is probably the oldest economic activity known to humanity.
One might view this problem has being composed of different interrelated parts. The first one is the definition
of the system. The second part is embedded in the word ‘manage’. This implies some action on the system.
The owner acts upon the resource through what one might call control mechanisms or policies. The last part is
disguised in the word ‘sustainably’. It is the goal to which the control aims.
In general, management of natural resources, such as forests or agricultural systems, involves manipulations of
the ecosystems to achieve goals such as a predictable flow of goods and services as food production, resource
extraction or conservation of particular ecological communities. In this context, Holling et al. (1996) examine
command and control management approaches, which often move ecosystems’ behaviors to a predetermined,
predictable state. Muradian (2001) states that human management of ecosystems is usually referred to as
control of ecological perturbations.
At this point, we may restate the initial problem as: what are the control policies that lead the bioeconomic
system to the goal proposed? So, the solution of the problem will be in some form of the optimal policies for
the controls.
Scientists, land managers, and others are proposing ecosystem management as the best way to manage natural
resources. If we are indeed moving toward ecosystem management as a new philosophy to guide resource
management, we must explore how this approach differs from the theoretical assumptions of current natural
resource management.
The challenge for theorists and policy analysts is to discern theoretical principles underlying current resource
management and to compare them to the principles proposed under ecosystem management. In this work we
follow a theoretical approach in the context of ecosystem management. We focus on the role played by the
ecosystem, and in particular its organization. The study of self-organization is rooted in the concept of
ecological integrity. Among other interpretations, it can be viewed as a guiding principle for sustainable land
1
Address: Avenida Rovisco Pais, 1, 1049-001 Lisboa, Portugal. Phone: +351 - 218 419 163, Fax: +351- 218
417 365, E-mail: [email protected]
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use that aims at long-term protection of ecological life-support systems (Kutsch et al., 2001). Effective use of
this concept in ecosystem sustainability requires theoretical integration in the context of resource management.
This paper is structured as follows. In the next section we present a conceptual discussion about the methods
and approaches to solve the self-organizing ecosystem management problem, and develop and present the
basic ecological model used in the bioeconomic system. In section 3 we present the main behaviours the
ecosystem can have depending on the ecosystem’s self-organizing capability. The optimal policies are then
presented and analysed in section 4. Section 5 concludes.
2. System definition and approaches
In this section we sketch some of the paradigms of ecosystem management. The concepts necessary for the
definition of the problem will be presented. We will start by defining the ecosystem as an object of research.
Then we will acknowledge the convenience of adopting this specific level of organisation for the analysis of
the sustainability of ecosystems within an economic framework. By the end of this section we will be able to
formulate the problem of sustainable management of the bioeconomic system.
2.1. Holistic vs. reductionist approaches
As mentioned above, we are dealing with the sustainability of ecosystems, in particular with the relation
between self-organisation and the sustainability of bioeconomic systems. Generally speaking, a bioeconomic
model consists of a biological (or biophysical) model that describes the behaviour of a living system, and an
economic model that relates the biological system to market prices and resource and institutional constraints.
The ecological system considered may range in level of aggregation, from populations to communities or
ecosystems. Therefore we need to be conscious about the level of organisation we choose to work with.
Generally, ecology may be approached from two perspectives; a reductionist one, where higher level
behaviour is nothing more than a definable combination of lower levels behaviours and a holistic one, where a
behaviour at one level cannot be explained in terms of lower level behaviours. Adopting a holistic philosophy,
the higher-level behaviour results from a synergism (Kay, 1983).
We take the perspective that ecology deals with irreducible systems. An ecosystem consists of so many
interacting components that it is impossible ever to be able to separate and examine all these relationships
(Kay and Schneider, 1994; Jorgensen 1992). This means that the ecosystem is not the sum of its parts and the
holistic perspective captures, in some way, the self-organisation of the ecosystem.
Our understanding of the dynamics of ecosystems, and consequently our ability to determine optimal
management policies would benefit greatly from the development of an overall paradigm for addressing the
self-organisation of living systems. For our purpose, we take self-organisation as describing the functional and
structural patterns of ecosystems. We must always remember that, left alone, living systems are selforganizing, that is, they will look after themselves (Kay, 1983). A damaged ecosystem, left to its own devices,
has the capability to regenerate if it has access to the information required for renewal, i.e. biodiversity, and if
the context for the information to be used, that is the biophysical environment, has not been altered (Kay and
Schneider, 1994).
2.2. Scale, aggregation and ecological production functions
Human interactions with nature have indirect effects on the functioning of the ecosystem as a whole on a longtime period. Harvesting practices are one important way through which humans change the composition and
structure of ecosystems to levels that may irreversibly affect their regeneration capacity and be a source of
ecological discontinuities (Muradian, 2001). Most radical changes occur when minimum viable biomass levels
are crossed, below which the extinction of the ecosystem is assured. These thresholds and bifurcation points
are crucial to the sustainability of societies and ecosystems and present a major problem for management
systems, since our knowledge of ecosystems’ dynamics near bifurcation points is limited (Rapport and
Whitford, 1999 in Hopkins, 2001).
Recognizing this, we assert that most perturbations are not reversible and not local, since indirect effects
spread in space and time in the ecological network. This brings the management problem to a long-term
ecosystem level of aggregation. In order to manage natural resources we must understand and address the
dynamics of populations and ecosystems at the organizational, temporal and spatial scales about which we
know least – large temporal and spatial scales (Pimm, 1991).
For ecosystem management, the ecological production function is the most useful feature of an ecological
model. The ecological production function or ecological productivity is simply the graph of dN/dt versus N,
being N the biomass. It is a highly aggregated representation of an ecosystem’s dynamics that depends on
innumerable biological and spatial details (Roughgarden, 1997). The ecological production function can be
thought of as indicating the interest or return on biomass products and services per unit time (Roughgarden,
1997). As a result, the return on biomass stocks can be compared to the return on financial stocks. We argue
that ecological production functions should include the dynamics of the structural organization of ecosystems
during ecological succession.
2.3. Ecosystem dynamics
We interpret the dynamics of biomass as following the dynamics of ecological succession. Odum (1969) notes
that ecological succession drives ecosystems’ biomass along a logistic curve and net production along a Uinverted curve. Following this, we assume that the regeneration function for biomass is given by the logistic
curve,
R( N )  rN (CC  N ) ,
(1)
where N is ecosystem biomass, rCC is the growth rate per unit biomass at low biomass values and CC is the
ecosystem carrying capacity. The important feature of ecological succession depicted by the logistic function
is the existence of a “reasonably directional development, which culminates in a stabilized ecosystem climax”
(Odum, 1969). Ecosystem carrying capacity is the value of biomass attained at this state, established by the
surrounding environment. It reflects the ecosystem’s organization that best allocates the available resources.
If a new unit of ecosystem biomass can alter the ecosystem carrying capacity, this may be understood as a
structural change, since an alteration of the efficiency of the allocation of resources has occurred – niche
specialization (Rodrigues et al., 2002). This interpretation relates the ecosystem carrying capacity to structural
properties of the system. Ecosystem carrying capacity can, this way, be viewed as a measure of the
organization of the ecosystem.
The basic assumption about a dynamic carrying capacity is that ecosystems while evolving organize
themselves into more complex systems, with more diversity and indirect interconnections between their
components at all levels, from species to communities (Mageau, 1998, Odum 1969; Kutsch et. al., 2001). For
instance, old-growth forests have attained a certain degree of organization approaching maturity, which allows
for services that rapid-growth forests cannot supply, like habitat for wildlife, repository of genetic diversity or
regulation of water flow or climate.
Accordingly, Rodrigues et al. (2002) proposed to consider carrying capacity as a dynamic state variable of the
ecosystem. The dynamic proposed is:
dCC
l dN
,

dt
N  h dt
(2)
and was adapted from a model devised by Cohen (1995) for human populations. The parameter l is the selforganizing capability of the ecosystem and h is the feedback saturation, since it imposes an upper limit to the
intensity of feedbacks between ecosystem biomass and ecosystem carrying capacity (Mota et al., 2004a).
Moreover, if l is high, it means that biomass has strong feedbacks with its carrying capacity and small changes
in population may have large effects on the ecosystem’s organisation. This way, we are endowing the
ecosystem with autocatalytic behaviour, since a rise in ecosystem biomass inflicts a rise in ecosystem carrying
capacity, which creates a greater potential for biomass to grow (provided that N < CC).
2.4. The bioeconomic system
We consider the economic system composed by the aggregate of all firms that directly extract resources from
the ecosystem and export a resource flow to the wider economy. We call this economic system the extractive
economy. This is an open economy that trades resources and investment flows with the wider economy for
investment rents and consumption goods as depicted in figure 2.1. We define such a system boundary in order
to separate the extractive sector from the productive (transformer and distributive) and service sectors,
therefore making it possible to highlight human perturbations on the ecological system in the context of
resource management.
The state of the extractive economy is described by its state variables: ecosystem biomass, N(t), and capital,
K(t). Decisions can be made at different levels, through the control variables: consumption, c(t), and rate of
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extraction of environmental products or harvesting rate, q(t). Once we choose to perform the optimisation at
the macroeconomic level, it makes sense to consider that the analysis is performed in continuous time and all
variables are continuous. This is because the discontinuities that result from the individual decisions of
harvesting or consumption will overlap, therefore being imperceptible at this scale.
Global Economy
K
Extractive Economy
V(q)
I
c
Figure 2.1 – Interactions between the extractive and global economy. The arrows in the figure represent
economic capital flows. V(q) is the revenue from exports of ecosystem biomass; K is the income from
investment in the global economy (with   0 , where  is the interest rate in the global economy) and c is
consumption expenditure. I denotes investment in accumulation of capital in the global economy.
The unperturbed ecosystem has a dynamics described in the previous section and further analysed in section 3.
Hence, the growth rate of the ecosystem is equal to the undisturbed regeneration function minus the extraction
rate:
dN
 R( N )  q .
dt
(3)
The extractive economy exports a flow of natural resources to the global economy with net economic revenue
V(q). This revenue depends on the supply of the ecosystem’s resources, q(t), since it is assumed that this flow
is sufficiently large to alter the price of the resources. The net economic revenue function, V(q), is considered
to have increasing returns but decreasing marginal returns to extraction rate.
This economy has two sources of income, one resulting from the exported natural resources flow, V(q), and
the other resulting from the interest obtained from the capital investments in the global economy, K, where
is the constant exogenous interest rate. This means that the extractive economy is small enough so that its
investments in the global economy have no effect on the interest rate.
The proceeds from these two sources of income can either be directed to consumption or invested in
accumulation of productive capital belonging to the extractive economy, dK/dt. Hence we have
dK
  K  V (q)  c .
dt
(4)
We assume that it is possible to define a utility function for this economy – citizens’ preferences are identical
and depicted by the representative consumer’s utility function, U(c) – and that a continuous and infinite
overlapping succession of individuals will behave rationally so as to maximise their utility function throughout
their lives. Hence, we are considering that future decisions feedback into the present to give an overall
maximum discounted sum of the consumers’ utility flow,


0
e tU (c)dt , where  is the discount or haste
rate. Population is assumed to be constant, so the problem is solved on the behalf of a representative agent.
3. 3 Ecosystem with structural dynamics
In this section we present in a very brief manner the main behaviours of an ecosystem whose structure is
changing along ecological succession. The results and figures have been derived in Mota et al. (2004).
We assume, as stated above, that the dynamics of the ecosystem is governed by (1) and (2). These equations
can be reduced to a single equation model (Mota et al., 2004):


dN
 N h
 rN  CCl  l ln 
  N   R( N ) .
dt
 b 


(5)
A simple conclusion drawn from the analogy with the logistic model is that this ecosystem has a carrying
capacity given by CCl  l ln  ( N  h) / b  which is constant for l  0 . That is, for zero self-organizing ability,
we have the conventional one-population dynamics (with no self-organization). So, the carrying capacity in
equation (5) is composed of an endogenous term l ln ( N  h) / b responsible for the autocatalytic reactions
in the ecosystem (feedbacks on carrying capacity) and a constant exogenous term CCl that depicts the
constraints of the surroundings. This way, the optimum operating point of the ecosystem is not imposed on the
system by exogenous conditions but depends on the historical information “remembered” by the system.
Feedbacks on carrying capacity may be positive or negative depending on the ecosystem biomass for a fixed
set parameters.
a)
b)
Figure 3.1 – a) Bifurcation diagram for the case where CCl  b. p) represents pure compensation; d) noncritical depensation and c) critical depensation. The line h=b is the asymptote of lcrit(1). b) Bifurcation
diagram for the case where b>CCl. e) represents unconditional extinction.
Depending on the feedbacks on carrying capacity, the ecosystem may have different kinds of behaviour. The
main conclusions are depicted in diagrams in the l-h plane, for fixed values of all the other parameters (Figure
3.1).
The line l  h separates pure compensation from depensation behaviours. In pure compensation, for all
natural capital values, each unit added to the existing stock causes a decrease in the per capita growth rate. In
depensation, for sufficiently low N, each unit added increases the per capita growth rate. If an ecosystem has
depensation behaviour with the additional property of negative growth rates for some biomass levels, it is said
to have critical depensation (Clark, 1976, p. 16). In this case, there is a certain threshold natural capital stock
level below which the ecosystem will become extinct. This can be viewed in figure 3.3 a). The curve l1 is a
pure compensation curve, the curve l2 is a depensation curve and the curve l3 represents a critical depensation
curve.
We conclude that if feedbacks on carrying capacity are low enough (small l) the ecosystem exhibits pure
compensation. On the other hand, if feedbacks on carrying capacity are sufficiently high the ecosystem
exhibits depensation and, for even higher feedbacks on carrying capacity, critical depensation. This reflects
the greater intensity of the relations among populations of the ecosystem: at equal low biomass levels,
ecosystems with higher self-organization ability have lower growth rates. This is due to the fact that an
ecosystem with higher self-organization has more requirements for maintenance. Hence, for low ecosystem
biomass its marginal growth is lower. This can be observed from figure 3.3.
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a)
b)
Figure 3.3 – a) Representation of the ecological production function given by equation (11) for different
values of l in the case of b-CCl<h<b. Respectively l1<h<l2<lcrit(1)<l3. b) Representation of the ecological
production function for different values of l in the case of h<b – CCl for the bifurcation diagram of figure
1 b).
Moreover, it is obvious from figure 3.3 that an ecosystem with high self-organization also has a higher realm
of extinction. This may be the case of the tropical and equatorial forests, which are strongly self-organised
ecosystems with numerous indirect effects and on the other hand are extremely sensitive to changes. What the
analysis of this model implies is that self-organizing ability may sometimes be detrimental to the ecosystem.
This implies that the best way to replenish an endangered ecosystem may sometimes not be to increase its
biodiversity or its self-organisation capability.
Considering a dynamic ecosystem carrying capacity as accounting for changes in the organisation of the
ecosystem, the model predicts a variety of behaviours for the ecosystem ranging from pure compensation to
depensation and critical depensation, within a single theoretical framework. In particular, it leads to the
prediction of the existence of threshold stock levels and bifurcation points, instead of building them in from
the outset.
4. Optimal management with structural dynamics
After briefly presenting the different kinds of behaviour the ecosystem here considered can exhibit, we discuss
the optimal management policies of an ecosystem with structural dynamics along ecological succession. In this
paper we will not focus on the mathematical details of the derivation of the optimal systems or phase portraits.
Rather, we are interested in presenting the optimal policies for the ecosystem with various kinds of selforganizing capability. For the mathematical details see Mota and Domingos (2004) where a fairly complete
analysis of this bioeconomic model is performed. Still, it is important the stress that we have given a special
treatment to the solutions where the extraction rate is zero. Consequently, we have found some special cases
of optimal polices that use zero extraction rates for some period of time. This kind of management policies is
seldom found in the body of work in this area.
4.1. The optimal management problem
Hence, the problem that this economy faces is to choose policies for the consumption c (t ) and the extraction
rates q (t ) in a context of interaction with the environment, in order to achieve the maximum inter-temporal
utility benefit. Formally, the problem is to:

max  U (c)e  t dt , subject to
c ,q
0
dN
 R( N )  q ,
dt
(6)
dK
  K  V (q)  c ,
dt
(7)
c, q, N, K > 0.
(8)
This is a usual problem of optimal control theory. For these types of problem there is a powerful tool –
Pontryagin’s maximum principle (see for instance Tu, 1994). Pontryagin’s maximum principle is a collection
of necessary conditions for optimality in the form of an optimal dynamical system for the state and control
variables. In the process are defined shadow prices for the various forms of stocks (in this case economic
capital and ecosystem biomass), which are the prices that produce the optimal exploitation of the resource if
used in real markets.
The dynamical system that allows for the derivation of the optimal management policies is:
dN
 R( N )  q ,
dt
(9)
dK
  K  V (q)  c ,
dt
(10)
1 dc   
,

c dt  (c)
(11)
1 dq V ' (q)

  R ' ( N )  ,
q dt V '' (q)
(12)
where  (c)  (c
U c )(dU c dc) is the elasticity of marginal utility relative to consumption.
Equation (11) gives the optimal path of the consumption rate while equation (12) is a version of the Hotelling
rule for efficient renewable resource extraction for an economy with market power in the resource, where the
resource is self-organizing along ecological succession. This condition simply states that any positive disparity
between the interest rate of the global economy and natural capital marginal productivity must be
compensated by an increase in the value of the stock, which is done by means of increasing the price of the
resource or reducing the flow of natural products to the global economy.
This system has 4 equations for 4 variables. However, the solution of (10) can be obtained after solving the
remaining equations. Additionally, it is possible to solve separately equation (11) and the system of equations
(9) and (12). Conceptually, this means that the problem of optimal management of natural capital is
independent of the optimal consumption problem for this society. For a formal proof of this proposition see
Mota and Domingos (2003).
This occurs because we do not consider nature’s services other than the supply of resources and it is assumed
that the extractive economy does not affect the global economy’s interest rate. In other words, this
independent maximisation of the wealth and of the revenue is mostly due to the fact that the ecosystem
resources and services only enter the economy (contributing to wealth) through the revenue function. As the
optimal consumption problem for this society is independent of the natural resource management problem we
may focus on the latter without loss of generality.
Hence, the system to be analysed here represents the optimisation of the extraction revenues,
dN
 R ( N )  q ( ) ,
dt
(13)
1 d
   R' ( N ) ,
 dt
(14)
where  is the shadow price of the ecosystem resources (Mota and Domingos, 2003)
In the next section we will describe the optimal policies for the management of an ecosystem considering three
different scenarios for the ecosystem – pure compensation, depensation and critical depensation.
4.2. Pure compensation
We present the results in phase portraits like, for instance figure 4.1, where the axes are the ecosystem
biomass, N, and the shadow price of the ecosystem biomass, . This representation is used since it allows the
7
corner solutions to be clearly depicted in the phase portrait (Mota and Domingos, 2003). Accordingly, the
shadow price (t) is inversely related to the extraction rate q(t). Moreover, we have zero extraction rates for
  V ' (0) .
In figure 4.1 we have the case of an ecosystem with pure compensatory behaviour.
Figure 4.1 – Phase diagram with the optimal trajectories for the revenue maximisation problem in the case of
an ecosystem with pure compensation behaviour.
The bioeconomic system possesses one optimal equilibrium solution N = N* which is a saddle point. The only
possibly optimal trajectories are the ones that have initial conditions on the two stable separatrices of the
saddle-point as depicted in figure 4.1. So, the optimal policy implies that the ecosystem will be harvested up
to the point, at which its marginal net productivity – which is the rate of return from holding the ecosystem – is
equal to the interest rate of the economy,   R ( N ) . This is the usual marginal productivity rule, now
generalised to account for feedback effects in the ecosystem. This rule as we will see cannot be used to derive
simple management policies in the case of depensation and critical depensation behaviours of the ecosystem.
'
a)
*
b)
Figure 4.2 – a) Bioeconomic equilibrium with an ecosystem with pure compensation behaviour. b) Optimal
stock paths N(t) with two different initial conditions.
In terms of management policies, the marginal productivity rule states that, if the interest rate exceeds the rate
of return from holding the ecosystem it is more profitable to harvest the resource at a rate higher than the
regeneration rate and invest the proceeds in the global economy. On the other hand, the ecosystem will
continue to grow or regenerate if that choice yields a greater net present value than immediate harvesting
without regeneration. Hence, a renewable resource is a relatively poor investment if its natural regeneration
rate is less than the rate of return of investing in capital in the global economy. Figure 4.2 shows the approach
to the bioeconomic equilibrium in the case of pure compensation. This solution corresponds to following the
separatrices in figure 4.1 b), and has the stock-level path depicted in figure 4.2 b).
Clark (1976) analysing a non-linear model of extraction of a renewable resource exhibiting pure compensation
behaviour states that “the optimal approach path is not the most-rapid approach; rather it is a more gradual
approach engendered by market reactions to the harvest rate”. The most rapid approach policy applied in the
linear theory is a special case of what engineers call a feedback control law. This term refers to control
policies (either optimal or non-optimal) with the property that the control is expressed directly as a function of
the state variable. An optimal feedback control law for the extraction of ecosystem products exhibiting pure
compensation with an infinite time horizon is obtained by numerically computing the separatrices show in
figure 4.1 b). This is the kind of laws we are interested in for the management of the ecosystem.
4.3. Depensation
If we are exploiting an ecosystem with depensation behavior, we expect that the optimal policy should be
different from that of an ecosystem with less self-organization (pure compensation). In particular, we would
expect that for low levels of the ecosystem’s biomass, since it is for low biomass levels that the main
qualitative differences appear in the marginal growth rates (see figure 3.3). In the case of depensation, as can
be seen in figure 4.3, there are now two equilibria, one stable and another unstable. Respectively, a saddlepoint and an unstable focus or node, depending on the interest rate.
Figure 4.3 – Bioeconomic equilibria
1  R ' ( N )
for an ecosystem with depensation behaviour. In this case
there is a zone of optimal extinction of the ecosystem
From figure 4.3 we can draw some general ideas about the optimal policy for the management of an ecosystem
*
with depensation behavior. If the ecosystem has a biomass higher than N 2 then the optimal solution would
be, as for an ecosystem with pure compensation, to harvest it at a rate higher than its regeneration.
Consequently the ecosystem biomass would be diminished, hence approaching it to the equilibrium
*
biomass N 2 . If the ecosystem has a higher marginal growth than the interest rate it is more profitable to
harvest the ecosystem at a rate lower than its regeneration, so that its biomass approaches the equilibrium
biomass. Even though this simple golden rule is widely accepted and used for renewable resource
management, here we see that it cannot be generalized to an ecosystem with depensation and critical
depensation behavior.
This can be seen in more detail from the respective phase portrait in figure 4.4. We conclude that the optimal
trajectories are those depicted in figure 4.4 b); the separatrices that converge asymptotically to the saddle
point and the trajectory that converges asymptotically to zero. So, if N 0  N 2 the optimal policy is to follow
*
the separatrix as in the case of pure compensation. If, on the other hand, N1  N 0  N 2 apparently there is
*
*
no single optimal solution in the neighborhood of the unstable focus, since there are several extraction rates
that puts the ecosystem in the optimal path.
This can also be seen from the biomass and extraction rate paths depicted in figure 4.5. In this figure we
depict the optimal biomass and extraction paths for the two different initial conditions considered above. Near
the focus the optimal path oscillates, making possible that different extraction rates be at the trajectory that
tends to the saddle point. The decision of what extraction rate to choose should then be made in terms of
revenue gains (i.e. choosing the global maximum from a set of local maxima).
9
In Mota and Domingos (2003) we prove the existence of a Skiba point near the unstable steady-state and
present a method for its numerical determination. A Skiba point is an initial state for which two different
optimal policies exist. In other words, in the Skiba point, it is indifferent what policy to follow. Moreover, the
existence of a Skiba point indicates that the long-term behavior of the bioeconomic system will be history
dependent: in our case, this means that according to the conditions prevailing in the first phases of
development, the ecosystem will converge to the saddle point or towards extinction. For initial states near the
unstable steady state small changes have dramatically different responses, one leading to the sustainable
conservation of the ecosystem and the other leading to its extinction.
Figure 4.4 – a) Phase diagram of the optimal resource management problem of an ecosystem exhibiting
depensation. b) Highlight of the optimal trajectories.
In the last case, if the optimal policy would be to extinguish the ecosystem a process of optimal extinction is
said to exist. This also happens in the case of an ecosystem with pure compensation if the interest rate is too
high. An ecosystem needs to be a high productivity system in order to be maintained by the owners. It is
important, however, to stress that the optimal extinction of an ecosystem involves costs that are not considered
in this model, such as increasing cost of extraction due to scarcity of resources, social costs associated with
environmental functions and other externalities. However it is a real policy option for the management of
natural resources.
a)
b)
Figure 4.5 – a) Biomass path. Two cases are shown: N(0)+>N2* and N1*<N(0)-<N2*. b) Extraction path . Two
cases are shown: q(0)+>q2* and q1*<q(0)-<q2*.
This should alert decision makers to study what type of ecosystem are they managing, whether it is an
ecosystem with pure compensation or depensation or critical depensation, and to be mindful of its sensitivity
to initial states when the ecosystem biomass is low.
Of course, the important conclusion is that for the management of an ecosystem with depensation behavior,
driving it to extinction may be an optimal management policy. Moreover, this only depends on the interest rate
and the initial level of the ecosystem biomass. If the interest rate has a sudden increase (more economic
pressure on the ecosystem), the realm of optimal extinction increases, as can be seen from figure 4.3 and 4.4.
However, both, pure compensation and depensation ecosystems may be prevented from optimal extinction if
the interest rate becomes sufficiently low. Moreover, if the ecosystem has depensatory behavior and if the
interest rate is low enough the optimal management policies will be the same as in the case of pure
compensation behavior.
4.4. Critical depensation
If the ecosystem has critical depensation behavior, the optimal system exhibits all of the phenomena of the
depensation model encountered above as well as an additional phenomenon: irreversibility. In the depensation
situation, if the ecosystem is on the path for optimal extinction, the manager can switch off extraction and the
ecosystem will regenerate, then he can again switch on the extraction. This is possible since in the zone of the
corner solution, for low biomasses there are trajectories that increase the biomass of the ecosystem. This may
seem to contradict the conclusion that the optimal extinction is a prediction of the model. However, the
optimal solutions predicted by the model are continuous in the state variables and the control. So, the model
could not predict the mentioned managing policy.
The optimal trajectories of the management problem are depicted in figure 4.7. The ecosystem threshold
biomass is where the dashed curve crosses the line 
 V ' (0) .
This is the zone of ecological driven extinction. All optimal paths that will at some time cross to this zone
were in the past driven by economic forces that could be halted by switching off the extraction and let the
ecosystem regenerate. Summing up, the economic driven extinction is not irreversible, however the ecological
driven extinction is.
Figure 4.7 – a) Phase diagram with the optimal trajectories for the optimal resource management problem of
an ecosystem exhibiting critical depensation.
As we have seen above, if the interest rate is low enough the optimal management policies for ecosystems with
depensation and pure compensation are the same. Notwithstanding, in managing an ecosystem with critical
depensation, lowering the interest rate will not make optimal extinction policies and history dependence
disappear. This can be seen if figure 4.8. In managing an ecosystem with critical depensation there are always
initial conditions for which it is optimal to extinguish the ecosystem, and there is always high history
dependence. This is an important result since we proved that only in special cases are the management policies
the same for ecosystems with different self-organization capability.
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Figure 4.8 – Bioeconomic equilibria with an ecosystem with depensation behaviour. In this case there is
always a zone of optimal extinction ecosystem.
Now, comparing two ecosystems with the same biomass, the conclusion, as expected, is that extracting and
exporting is still a good investment for higher interest rates or, for the same interest rate, the equilibrium
population corresponding to the saddle point is greater. This can be seen from figure 4.9, where we plotted the
marginal growth curves for two ecosystems, one with pure compensation and the other with depensation
behaviour. This means that when the system loses organization it also looses productivity and competitiveness
for the global economy. The ecosystem’s self-organization potentiates the creation of new ecosystem biomass,
increasing the surplus from holding the resources, and consequently augmenting the range for economic
profitability. On the other hand, for low biomass levels the ecosystem with pure compensation behaviour has a
greater surplus. This can be understood in the light of the interpretation we used in section 3. The ecological
production function of the ecosystem can be interpreted as the difference between natural resource
reproduction and resource use for maintenance. Accordingly, for low biomass levels, a high self-organization
ecosystem has high maintenance requirements. Thus, the ecosystem has lower marginal growth rates.
There are, then, two effects that need to be balanced in decision making. The managers need to balance the
need for high productivity of the ecosystem with its high risk of extinction for low ecosystem biomass. The
existence of these two effects is due to autocatalytic relations between the actual biomass and ecosystem
carrying capacity that potentiates extreme states of productivity for the ecosystem, whether they are of very
high productivity or very low.
Figure 4.9 – Comparison of ecosystems with different self-organization capability. In this case, as can be seen
from figure 3.1a) we have, l1 < lcrit(1) < l2.
5. Conclusions
In this work we have discussed the relevance of ecosystems’ self-organization along ecological succession in
the context of optimal resource management. So, the model of optimal management of ecosystems we
presented internalises the effects of self-organization in the management policies.
For this, we use a phenomenological model for the ecosystem dynamics that grasps the dependence of the
ecosystem productivity on its self-organisation. This model was based on the model of Cohen (1995),
originally devised for human population dynamics, where population growth allows the population to expand
its carrying capacity. Here we analyse it in a more ecological perspective and incorporate it in a bioeconomic
model for optimal ecosystem management.
With the specific relations considered for ecosystem biomass and ecosystem carrying capacity, this model
describes human-nature interactions that affect the carrying capacity of the ecosystem. The alteration of the
ecosystem carrying capacity is assumed to be a measure of the self-organisation of the system. This has a
sound ecological interpretation, related to niche specialisation (section 2.3) along ecological succession.
We then showed how the model described could be used to derive an ecological production function that
incorporates a density dependent carrying capacity. Using this ecological production function we concluded
that if the feedbacks on carrying capacity are low enough the ecosystem exhibits pure compensation. On the
other hand, if the feedbacks on carrying capacity are sufficiently high the ecosystem exhibits depensation and
critical depensation, reflecting the greater intensity of relations among species of the ecosystem (Section 3).
Hence we produced a single theoretical framework that captures different qualitative behaviours of the
ecosystem and relates them to its self-organisation. Moreover, the threshold values for ecosystem biomass are
endogenous to the model.
Next, we devised a bioeconomic model where the ecosystem has the regeneration function discussed above.
We performed a dynamical optimisation of an extractive economy exploiting a self-organizing ecosystem. The
main conclusion is that the optimal policies will, generally, depend on the dynamics of the ecosystem. Only in
the case of the interest rate being low enough are the pure compensation and depensation ecosystems managed
with the same type of optimal policies.
So we have three different sets of optimal solutions depending on whether the ecosystem has pure
compensation, depensation or critical depensation behaviours. For pure compensation there are two optimal
trajectories. For depensation there are three optimal trajectories. In this case, there exists a possibility of
optimal extinction of the ecosystem. The interesting effect in an ecosystem with depensation behaviour is that
there is a point where there are two optimal policies – Skiba point. Hence, there is history dependence in this
bioeconomic model. Near the unstable steady-state small changes in initial conditions can put the ecosystem
on the optimal extinction path or on the saddle point path.
In the case of an ecosystem with critical depensation behaviour we encountered an additional phenomenon:
irreversibility. In the critical depensation behaviour if the ecosystem biomass is lower than the threshold
biomass the ecosystem will be driven to extinction no matter what happens to future extraction rates.
Generally, we can conclude that an ecosystem with high self-organisation capability has a wider range of
profitability than an ecosystem with lower self-organisation capability (Figure 4.9). On the other hand, it also
has a higher realm of optimal extinction. So, the managers must balance the need for high productivity of the
ecosystem with its high risk of extinction.
Though the analysis presented here is mainly theoretical, the objective of devising optimal ecosystem
management models is to derive policies useful for the decision makers. The main purpose of this work was to
bring into the language of ecosystem management the concept of self-organisation of ecosystems with a clear
mathematical formulation making self-organisation an operational concept.
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